Open and Distance Learning

Effects

Before considering the procedures for estimating parameters in the models, let us explore further the nature of the association in an r x c table. The aim consists in testing the presence of particular effects which affect the frequencies of the cross-classified variables in an r x c table [Brown 1974, Brown 1976].

In this process, that according to Brown we call screening effects, we will consider:

a)

a general effect;

b)

main effects due to single variables;

c)

an effect due to the interaction of the two variables.

Table 2.3: Effects, null hypotheses, estimated expected frequencies, degrees of freedom

Effects	Null Hypotheses	Est. exp. frequencies	d.f.
General effect	p11 = ¼ = prc = [1/(r c)]	n11 = ¼ = nrc = [N/(rc)]	rc-1
Single Factor Effects	p1. = ¼ = pr. = 1/r p.1 = ¼ = p.c = 1/c	n1. = ¼ = nr. = N/r n.1 = ¼ = n.c = N/c	r - 1 c - 1
Interact. Effect	pij = pi. p.j	[(ni. n.j)/(N)]	(r-1)(c-1)

In table 2.3, for general effect, the null hypothesis specifies the equal probability of all the r x c cells, hence the equidistribution of N in the r x c cells. For each single variable (factor), the null hypothesis specifies the equal probability of the categories, hence the equidistribution of N in the categories. For the interaction effect, the null hypothesis specifies that the two variables X and Y are independent. The likelihood ratio statistic G² is used to test the hypotheses in table 2.3.

Table 2.4: Effects, symbols, likelihood ratio statistics, degrees of freedom

Effects	Sym.	Likelihood ratio statistics	d.f.
General Effect	-	Gg2 = 2{årc nij log ( nij / [N/rc] )}	rc-1
Single Factor Effects	X Y	GX2 = 2{år ni. log ( ni. / N/r )} GY2 = 2{åc n.j log ( n.j / N/c )}	r-1 c-1
Interact. Effect	XY	GXY2 = 2{år åc nij log ( nij /[(ni. n.j)/N] )}	(r-1)(c-1)

Note. Effect symbols do not have brackets.
A high value of the G² statistic (p < .05) provides evidence that the null hypothesis is false and that there is an interpretable effect due to a particular factor (variable) or to the interaction of X and Y.

In table 2.4 the G² effect statistics are presented. The general effect G_g² includes the single factor effects and the interaction effect. That is,

G_g² = G_X² + G_Y² +G_XY²,

hence

G_XY² = G_g² - (G_X² + G_Y²);

G_X² = G_g² - (G_XY² + G_Y²);

G_Y² = G_g² - (G_XY² + G_X²);

that is, the three effects can be obtained also by difference. Similarly, for the degrees of freedom,

(rc-1) = (r-1) + (c-1) + (r-1)(c-1),

hence,

(r-1)(c-1) = (rc-1) - [(r-1) + (c-1)]

(r-1) = (rc-1) - [(r-1)(c-1) + (c-1)]

(c-1) = (rc-1) - [(r-1)(c-1) + (r-1)]

Considering the degrees of freedom we observe that the only constraint in terms of cell frequencies is å^r å^c n_ij = N, so that there are (r c - 1) linearly independent cell frequencies in the table.

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